Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-07T08:45:54.903Z Has data issue: false hasContentIssue false
  • English
  • Français

Pairwise Balanced Designs with Block Sizes Three and Four

Published online by Cambridge University Press:  20 November 2018

Charles J. Colbourn ,
Alexander Rosa  and
Douglas R. Stinson
Charles J. Colbourn
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1
Alexander Rosa
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1
Douglas R. Stinson
Affiliation:
Department of Computer Science and Engineering, University of Nebraska - Lincoln, Lincoln, Nebraska 68858, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given integers ν, a and b, when does a pairwise balanced design on ν elements with a triples and b quadruples exist? Necessary conditions are developed, and shown to be sufficient for all v ≥ 96. An extensive set of constructions for pairwise balanced designs is used to obtain the result.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 4 , 01 August 1991 , pp. 673 - 704
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Assaf, A. and Hartman, A., Resolvable group divisible designs with block size 3, Discrete Mathematics 77(1989), 5–20.Google Scholar
2. Batten, L.M. and Totten, J., On a class of linear spaces with two consecutive line degrees, Ars Combinatoria 10(1980), 107–114.Google Scholar
3. Beth, T., Jungnickel, D. and Lenz, H., Design Theory. Cambridge University Press, Cambridge, 1986.Google Scholar
4. Bose, R.C., Shrikhande, S.S. and Parker, E.T., Further results on the construction of mutually orthogonal latin squares and the falsity ofEuler's conjecture, Canadian Journal of Mathematics 12(1960), 189–203.Google Scholar
5. Brouwer, A.E., Optimal packings of A4 's into a Kn, Journal of Combinatorial Theory Series A 26(1979), 278–297.Google Scholar
6. Brouwer, A.E., The linear spaces on 15 points, Ars Combinatoria 12(1981), 3–35.Google Scholar
7. Brouwer, A.E., Hanani, H. and Schrijver, A., Group divisible designs with block size four, Discrete Mathematics 20(1977), 1–10.Google Scholar
8. Colbourn, C.J., Hoffman, D.G. and Rees, R., Group-divisible designs with block size three, Research Report M/CS 89–24. Mount Allison University, 1989.Google Scholar
9. Colbourn, C.J., Pulleyblank, W.R. and Rosa, A., Hybrid triple systems and cubic feedback sets, Graphs and Combinatorics 5(1989), 15–28.Google Scholar
10. Colbourn, C.J. and Rôdl, V., Percentages in pairwise balanced designs, Discrete Mathematics 77(1989), 57–63.Google Scholar
11. Doyen, J. and Wilson, R.M., Embeddings ofSteiner triple systems, Discrete Mathematics 5(1973), 229–239.Google Scholar
12. Hanani, H., The existence and construction of balanced incomplete block designs, Annals of Mathematical Statistics 32(1961), 361–386.Google Scholar
13. Heinrich, K. and Zhu, L., Existence of orthogonal Latin squares with aligned subsquares, Discrete Mathematics 59(1986), 69–78.Google Scholar
14. Kelly, L.M. and Nwankpa, S., Affine embeddings of Sylvester-Gallai designs, Journal of Combinatorial Theory Series A 14(1973), 422–438.Google Scholar
15. Kirkman, T.P, On a problem in combinations, Cambridge and Dublin Mathematical Journal 2(1847), 191— 204.Google Scholar
16. Lindner, C.C. and Rosa, A., Steiner triple systems having a prescribed number of triples in common, Canadian Journal of Mathematics 27(1975), 1166–1175. Corrigendum: 30(1978), 896.Google Scholar
17. Mills, H., On the covering of pairs by quadruples II, Journal of Combinatorial Theory Series A 15(1973), 138–166.Google Scholar
18. Raghavarao, D., Constructions and Combinatorial Problems in the Design of Experiments, (updated edition), Dover Publications, Mineola NY, 1988.Google Scholar
19. Rees, R., Uniformly resolvable pairwise balanced designs with block sizes two and three, Journal of Combinatorial Theory Series A 45(1987), 207–225.Google Scholar
20. Rees, R., The existence of restricted resolvable designs I: ( 1,2)-factorizations of Kin , Discrete Mathematics, to appear.Google Scholar
21. Rees, R., The existence of restricted resolvable designs II: (1,2)-factorizations ofKin+x, Discrete Mathematics, to appear.Google Scholar
22. Rees, R., The spectrum of restricted resolvable designs with r = 2, IMA Preprint Series #538, Institute for Mathematics and Its Applications, University of Minnesota, 1989.Google Scholar
23. Rees, R. and Stinson, D.R., On the existence of incomplete designs of block size four having one hole, Utilitas Mathematica 35(1989), 119–152.Google Scholar
24. Rees, R. and Stinson, D.R., On resolvable group divisible designs of block size 3, Ars Combinatoria 23(1987), 107–120.Google Scholar
25. Rees, R. and Stinson, D.R., Kirkman triple systems with maximum subsystems, Ars Combinatoria 25(1988), 125–132.Google Scholar
26. Rees, R. and Stinson, D.R., On combinatorial designs with subdesigns, to appear.Google Scholar
27. Rosa, A. and Hoffman, D.G., The number of repeated blocks in twofold triple systems, Journal of Combinatorial Theory Series A 41(1986), 61–88.Google Scholar
28. Spencer, J., Maximal consistent families of triples, Journal of Combinatorial Theory 5(1968), 1–8.Google Scholar
29. Stanton, R.G., The exact covering of pairs on nineteen points with block sizes two, three and four, Journal of Combinatorial Mathematics and Combinatorial Computing 4(1988), 69–11.Google Scholar
30. Stanton, R.G. and Allston, J.L., A census of values for g(k)(, 1,2; v), Ars Combinatoria 20(1985), 203–216.Google Scholar
31. Stern, G. and Lenz, H., Steiner triple systems with given subsystems: another proof of the Doyen-Wilson theorem, Bolletino UMI A 5(1980), 109–114.Google Scholar
32. Stinson, D.R., Hill-climbing algorithms for the construction of combinatorial designs, Annals of Discrete Mathematics 26(1985), 321–334.Google Scholar
33. A.R Street and Street, D.J., Combinatorics of Experimental Design. Oxford University Press, Oxford and New York, 1987.Google Scholar
34. Todorov, D.T., Three mutually orthogonal latin squares of order 14, Ars Combinatoria 20(1985), 45–47.Google Scholar
35. Wilson, R.M., Constructions and uses ofpairwise balanced designs, Math. Centre Tracts 55(1974), 18–41.Google Scholar